A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.

A stabilized conforming nodal integration for Galerkin mesh-free methods

https://orbilu.uni.lu/bitstream/10993/13726/1/phu-meshless.pdf

Review: Meshless methods: A review and computer implementation aspects

A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

Recent developments of meshfree and particle methods and their applications in applied mechanics are surveyed. Three major methodologies have been reviewed. First, smoothed particle hydrodynamics ~SPH! is discussed as a representative of a non-local kernel, strong form collocation approach. Second, mesh-free Galerkin methods, which have been an active research area in recent years, are reviewed. Third, some applications of molecular dynamics ~MD! in applied mechanics are discussed. The emphases of this survey are placed on simulations of finite deformations, fracture, strain localization of solids; incompressible as well as compressible flows; and applications of multiscale methods and nano-scale mechanics. This review article includes 397 references. @DOI: 10.1115/1.1431547#

Meshfree and particle methods and their applications

The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

A new meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation, is proposed in the present paper. The total formulation and a rate formulation are developed for the implementation of the present method. The present method does not need domain and boundary elements to deal with the volume and boundary integrals, which will cause some difficulties for the conventional boundary element method (BEM) or the field/boundary element method (FBEM), as the volume integrals are inevitable in dealing with nonlinear boundary value problems. This is the same for the element free Galerkin (EFG) method which also needs element-like cells in the entire domain to evaluate volume integrals. The “companion fundamental solution” introduced in Zhu, Zhang and Atluri (1998) is used so that no derivatives of the shape functions are needed to construct the stiffness matrix for the interior nodes, as well as for those nodes with no parts of their local boundaries coinciding with the global boundary of the domain of the problem, where essential boundary conditions are specified. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very efficient, in the present nonlinear LBIE approach. Numerical examples are presented for several problems, for which exact solutions are available. The present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives. No post processing procedure is required to compute the derivatives of the unknown variable (as in the conventional FBEM), since the solution from the present method, using the moving least squares approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence for the Sobolev norms ∥·∥0 and ∥·∥1 are achievable, and that the values of the unknown variable and its derivatives are quite accurate.

T. Zhu

Papers

A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

A meshless local boundary integral equation (LBIE) method for solving nonlinear problems